1969, 1973, 1985, 1993, 1997, 1998 AP Multiple Choice Sections, AB and BC, Solutions (620)

X 2 dx a 4 e1 x c d b 1 c x 1 for all

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Unformatted text preview: ote: In this examination, ln x denotes the natural logarithm of x (that is, logarithm to the base e). 1. If f ( x) = e1 x , then f ′( x) = (A) 2. e1 x x 3 (B) −e1 x 2 ∫ 0 ( x + 1) (A) 3. − 12 (C) e1 x x (C) 16 3 21 2 If f ( x) = x + (B) 7 (D) ( −∞, − 1] ∪ [1, ∞ ) (B) 14 3 [ −1,1] ( 0, ∞ ) (E) 1 (1 x )−1 e x (E) − ( −∞, ∞ ) 1 4 ( −∞, 0 ) ∪ ( 0, ∞ ) 1 For what non-negative value of b is the line given by y = − x + b normal to the curve y = x3 ? 3 4 10 10 3 (A) 0 (B) 1 (C) (D) (E) 3 3 3 2 ∫ −1 x dx is x (A) –3 6. (E) 1 , then the set of values for which f increases is x (D) 5. x 2 dx = (A) 4. e1 x (C) (D) (B) 1 (C) x −1 for all x ≠ −1, then f ′(1) = x +1 1 (A) –1 (B) − (C) 2 2 (D) 3 0 (D) (E) nonexistent If f ( x) = AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 1 2 (E) 1 29 1973 AP Calculus BC: Section I 7. ( ) If y = ln x 2 + y 2 , then the value of (A) 0 8. 1 2 (B) dy at the point (1, 0) is dx (C) 1 (D) 2 (E) undefined If y = sin x and y ( n ) means “the nth derivative of y with respect to x,” then the smallest positive integer n for which y ( n ) = y is (A) 2 9. (B) 4 If y = cos 2 3 x , then (C) 5 (D) 6 (E) 8 dy = dx (A) −6 sin 3 x cos 3 x (B) −2 cos 3 x (D) 6 cos 3 x (E) 2 sin 3 x cos 3 x (C) 10. The length of the curve y = ln sec x from x = 0 to x = b, where 0 < b < 2 cos 3 x π , may be expressed by 2 which of the following integrals? (A) b ∫ 0 sec x dx b (B) ∫ 0 sec (C) ∫0 (D) ∫0 (E) ∫0 b 2 x dx (sec x tan x) dx b 1 + ( ln sec x ) dx b 1 + sec 2 x tan 2 x dx 2 ( ) 11. Let y = x 1 + x 2 . When x = 0 and dx = 2, the value of dy is (A) –2 (B) –1 (C) 0 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) 1 (E) 2 30 1973 AP Calculus BC: Section I 12. If n is a known positive integer, for what value of k is k n −1 ∫1 x dx = 1 ? n 1n (B) (E) (A) 0 (D) 21 n 1n ⎛2⎞ ⎜⎟ ⎝n⎠ ⎛ 2n − 1 ⎞ ⎜ ⎟ ⎝n⎠ 2n (C) 13. The acceleration α of a body moving in a straight line is given in terms of time t by α = 8 − 6t . If the velocity of the body is 25 at t = 1 and if s (t ) is the distance of the body from the origin at time t, what is s (4) − s (2) ? (A) 20 (B) 24 14. If x = t 2 − 1 and y = 2et , then (A) et t 2et t (B) (C) 28 (D) 32 (E) 42 dy = dx (C) e t t 2 (D) 4et 2t − 1 (E) et 15. The area of the region bounded by the lines x = 0, x = 2, and y = 0 and the curve y = e x 2 is e −1 2 (B) e −1 16. A series expansion of 2 ( e − 1) sin t is t (A) (A) 1 − (C) (D) 2e − 1 (E) 2e t2 t4 t6 +−+ 3! 5! 7! (B) 1 t t3 t5 −+−+ t 2! 4! 6! (C) 1+ (D) 1 t t3 t5 ++++ t 2! 4! 6! (E) t− t2 t4 t6 +++ 3! 5! 7! t3 t5 t7 +−+ 3! 5! 7! AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. 31 1973 AP Calculus BC: Section I 17. The number of bacteria in a culture is growing at a rate of 3, 000e 2t 5 per unit of time t. At t = 0 , the number of bacteria present was 7,500. Find the number present at t = 5 . (A) 1, 200e 2 (B) 3, 000e 2 (C) 7,500e 2 (D) 7,500e5 15, 000 7 e 7 (E) 18. Let g be a continuous function on the closed interval [ 0,1] . Let g (0) = 1 and g (1) = 0 . Which of the following is NOT necessarily true? (A) There exists a number h in [ 0,1] such that g (h) ≥ g ( x) for all x in [ 0,1] . (B) For all a and b in [ 0,1] , if a = b , then g (a) = g (b) . 1 . 2 3 (D) There exists a number h in [ 0,1] such that g (h) = . 2 (C) There exists a number h in [ 0,1] such that g (h) = (E) For all h in the open interval ( 0,1) , lim g ( x) = g (h) . x →h 19. Which of the following series converge? ∞ ∑ I. n =1 1 n2 (A) I only 20. ∫x II. ∞ ∑ n =1 (B) III only 1 n (D) 21. (C) I and II only (−1)n n (D) I and III only (E) I, II, and III 4 − x 2 dx = ( 4 − x2 ) − ( x2 4 − x2 1 e3 2 ) x2 +2 x ) − 4− x (E) ( 4 − x2 ) − 32 +C (C) ( x2 4 − x2 ) 3 32 +C 32 +C 3 ( 2 32 (B) +C 3 ∫ 0 ( x + 1) e (A) ∑ n =1 32 (A) ∞ III. 3 +C dx = (B) e3 − 1 2 (C) e4 − e 2 AP Calculus Multiple-Choice Question Collection Copyright © 2005 by College Board. All rights reserved. Available at apcentral.collegeboard.com. (D) e3 − 1 (E) e4 − e 32 1973 AP Calculus BC: Section I 22. A particle moves on the curve y = ln x so that the x-component has velocity x′(t ) = t + 1 for t ≥ 0 . At time t = 0 , the particle is at the point (1, 0 ) . At time t = 1 , the particle is at the point (A) (B) ( e2 , 2 ) (D) 23. ( 2, ln 2 ) ( 3, ln 3) (E) 3⎞ ⎛3 ⎜ , ln ⎟ 2⎠ ⎝2 (C) 1 2 (C) 5⎞ ⎛5 ⎜ , ln ⎟ 2⎠ ⎝2 1 ⎛ 2+h⎞ ln ⎜ ⎟ is h →0 h ⎝ 2 ⎠ lim (A) e2 (B) 1 (D) 0 (E) nonexistent 24. Let f ( x) = 3x + 1 for all real x and let ε > 0 . For which of the following choices of δ is f ( x) − 7 < ε whene...
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This note was uploaded on 12/29/2010 for the course MATH 214 taught by Professor Smith during the Fall '10 term at Oregon Tech.

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